Is it permissible for a girl to call her teacher "my mother" if she trusts her greatly and considers her the closest person to her? And could this intense love for her reach the point of attachment forbidden by Sharia, especially if her actions are influenced by seeking to please her? And what is the solution for that?

There is no harm in a student addressing her female teacher as "Ya Ummi" (Oh, my mother) if it is done out of respect and affection. Furthermore, loving a teacher for the sake of Allah does not detract from one's faith; rather, it is one of its signs, provided that one is mindful of intention, moderation, and avoids exaggeration.